In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using net primary productivity (NPP).
Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: Productivity ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: Productivity ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.
The relationship between Shannon diversity and productivity was qualitatively similar to that of Shannon diversity for the majority of the models (4/6).
In Forest2, while the within-treatment slopes are consistent between biomass and productivity, the across-treatment slopes differ. During the with stage, the across-treatment slope is insignificant with biomass, but significant and slightly positive for productivity. During the without seed rain phase, the significant and negative across-treatment slope of the relationship between biomass and Shannon diversity becomes insignificant for productivity.
Dryland displays the most variation between the two measures of ecosystem functioning. The predominant difference is shows in the within-treatment slopes, as they flip from being significantly positive to significantly negative in both the with and without seed rain phases. Secondly, while the across-treatment slope is insignificant during the without seed rain phase for biomass as the ecosystem functioning, for productivity the slope is significant and positive. The reason for this change is clerical, because while seed biomass is incorporated into the productivity calculations, it is left absent from the total biomass calculations.
Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.
Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on the focal ecosystem function (either productivity or biomass), we find that the aggregate patterns are nearly identical between ecosystem functions.
This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and productivity as the focal ecosystem function.
Important terms:
Stage: With seed rain, without seed rainNinitial: Planted species richnessClark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 2.41 3.65 -4.92 9.36 1.00 2004 1992
## StageWithoutseedrain -12.32 3.97 -20.18 -4.43 1.00 2208 1822
## StageWithseedrain:Shannon 17.73 1.52 14.85 20.79 1.00 2006 2205
## StageWithoutseedrain:Shannon 25.76 1.93 22.03 29.64 1.00 2217 2080
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 23.19 0.59 22.09 24.40 1.00 2991 2223
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2964833 0.02290795 0.2506286 0.3416683
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 82.98 26.46 32.04 133.75 1.00 5142 2639
## Ninitial4:StageWithseedrain 151.80 23.75 105.63 197.65 1.00 5072 2558
## Ninitial8:StageWithseedrain 166.37 20.19 125.25 205.31 1.00 5075 2456
## Ninitial16:StageWithseedrain 250.82 17.22 217.24 284.75 1.00 4946 2680
## Ninitial32:StageWithseedrain 278.86 19.96 240.30 319.25 1.00 5166 2906
## Ninitial2:StageWithoutseedrain 23.07 10.45 2.87 43.20 1.00 5316 2681
## Ninitial4:StageWithoutseedrain 64.50 14.48 36.02 93.12 1.00 5568 2782
## Ninitial8:StageWithoutseedrain 89.16 14.45 60.45 116.74 1.00 6188 2872
## Ninitial16:StageWithoutseedrain 184.60 18.03 148.68 219.84 1.00 5397 2686
## Ninitial32:StageWithoutseedrain 188.80 23.85 142.65 235.24 1.00 5119 2636
## Ninitial2:StageWithseedrain:Shannon -36.63 16.47 -68.11 -4.97 1.00 5162 2743
## Ninitial4:StageWithseedrain:Shannon -54.41 10.90 -75.70 -33.16 1.00 5090 2557
## Ninitial8:StageWithseedrain:Shannon -47.37 7.60 -62.11 -32.12 1.00 5168 2396
## Ninitial16:StageWithseedrain:Shannon -65.00 5.84 -76.68 -53.50 1.00 4913 2813
## Ninitial32:StageWithseedrain:Shannon -64.85 6.32 -77.59 -52.75 1.00 5045 2980
## Ninitial2:StageWithoutseedrain:Shannon -3.40 7.02 -17.33 10.27 1.00 5415 2744
## Ninitial4:StageWithoutseedrain:Shannon -19.28 7.44 -33.91 -4.67 1.00 5533 2968
## Ninitial8:StageWithoutseedrain:Shannon -23.94 6.41 -36.08 -11.31 1.00 6048 2895
## Ninitial16:StageWithoutseedrain:Shannon -52.70 7.42 -67.30 -37.91 1.00 5485 2624
## Ninitial32:StageWithoutseedrain:Shannon -44.05 9.10 -61.90 -26.51 1.00 5236 2616
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.05 0.43 14.22 15.91 1.00 6987 2815
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.6995809 0.01090739 0.6758253 0.7192806
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 44.48 1.56 41.44 47.61 1.00 2172 2350
## StageWithoutseedrain 37.51 1.75 34.12 40.93 1.00 1867 1747
## StageWithseedrain:Shannon 8.72 0.55 7.63 9.80 1.00 2231 2550
## StageWithoutseedrain:Shannon 15.09 0.77 13.52 16.58 1.00 1826 2017
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 12.22 0.31 11.64 12.86 1.00 2838 2542
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4418439 0.01928773 0.4022911 0.4776298
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 85.32 32.40 22.50 147.16 1.00 4683 2998
## Ninitial4:StageWithseedrain 136.68 34.69 69.60 206.52 1.00 4808 2845
## Ninitial8:StageWithseedrain 133.48 32.48 68.65 198.41 1.00 5798 2701
## Ninitial16:StageWithseedrain 92.80 52.95 -12.42 197.98 1.00 4724 2071
## Ninitial32:StageWithseedrain 45.14 61.91 -78.54 168.23 1.00 4742 2505
## Ninitial2:StageWithoutseedrain -16.38 22.60 -59.62 27.96 1.00 5166 2750
## Ninitial4:StageWithoutseedrain 27.19 13.72 -0.00 53.15 1.00 5140 3046
## Ninitial8:StageWithoutseedrain 38.70 10.52 18.48 60.01 1.00 4959 2496
## Ninitial16:StageWithoutseedrain 47.64 11.54 24.61 69.88 1.00 5143 2510
## Ninitial32:StageWithoutseedrain 72.50 14.94 43.38 101.49 1.00 5581 2919
## Ninitial2:StageWithseedrain:Shannon -16.34 19.63 -54.07 21.95 1.00 4677 2986
## Ninitial4:StageWithseedrain:Shannon -30.08 15.04 -60.46 -1.01 1.00 4833 2802
## Ninitial8:StageWithseedrain:Shannon -20.22 11.21 -42.57 2.11 1.00 5820 2790
## Ninitial16:StageWithseedrain:Shannon -4.79 14.93 -34.52 24.99 1.00 4707 2001
## Ninitial32:StageWithseedrain:Shannon 7.60 14.75 -21.65 37.07 1.00 4745 2408
## Ninitial2:StageWithoutseedrain:Shannon 45.98 13.98 18.37 72.55 1.00 5202 2542
## Ninitial4:StageWithoutseedrain:Shannon 22.09 7.44 7.96 36.93 1.00 5147 3020
## Ninitial8:StageWithoutseedrain:Shannon 17.69 4.76 7.96 26.96 1.00 5024 2553
## Ninitial16:StageWithoutseedrain:Shannon 12.15 4.22 4.05 20.49 1.00 5188 2645
## Ninitial32:StageWithoutseedrain:Shannon 3.28 4.45 -5.50 11.98 1.00 5562 2804
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.04 0.25 8.56 9.55 1.00 7003 3231
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4944095 0.01970995 0.4547698 0.530752
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 46.11 1.42 43.42 48.97 1.00 2118 2043
## StageWithoutseedrain 46.09 1.43 43.24 48.81 1.00 1781 1908
## StageWithseedrain:Shannon 2.69 0.53 1.62 3.71 1.00 2093 2070
## StageWithoutseedrain:Shannon 2.84 0.58 1.73 4.00 1.00 1831 1757
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.96 0.25 9.49 10.46 1.00 3213 2205
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.06514213 0.01638124 0.03489941 0.09834952
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 17.56 37.59 -56.23 90.86 1.00 3080 2756
## Ninitial4:StageWithseedrain -7.56 25.50 -56.56 42.17 1.00 3610 2794
## Ninitial8:StageWithseedrain 4.11 25.57 -46.14 54.75 1.00 3611 2851
## Ninitial16:StageWithseedrain 58.76 23.15 12.86 102.98 1.00 3659 2877
## Ninitial32:StageWithseedrain 47.83 30.40 -10.57 106.73 1.00 3799 2901
## Ninitial2:StageWithoutseedrain 64.81 6.41 52.61 77.39 1.00 3175 2584
## Ninitial4:StageWithoutseedrain 25.46 10.98 3.20 46.72 1.00 3587 2592
## Ninitial8:StageWithoutseedrain 37.27 11.66 14.85 60.21 1.00 3429 2875
## Ninitial16:StageWithoutseedrain 64.81 21.39 24.68 107.89 1.00 3321 2812
## Ninitial32:StageWithoutseedrain 78.13 32.48 10.28 140.64 1.00 3474 2677
## Ninitial2:StageWithseedrain:Shannon 18.79 22.51 -25.01 62.97 1.00 3070 2721
## Ninitial4:StageWithseedrain:Shannon 25.33 11.16 3.49 46.73 1.00 3626 2821
## Ninitial8:StageWithseedrain:Shannon 17.35 9.05 -0.62 35.14 1.00 3617 2896
## Ninitial16:StageWithseedrain:Shannon -1.04 6.86 -14.27 12.53 1.00 3658 2855
## Ninitial32:StageWithseedrain:Shannon 2.62 7.99 -12.86 17.96 1.00 3797 2985
## Ninitial2:StageWithoutseedrain:Shannon -9.93 3.97 -17.78 -2.21 1.00 3164 2639
## Ninitial4:StageWithoutseedrain:Shannon 11.62 5.12 1.81 21.97 1.00 3581 2706
## Ninitial8:StageWithoutseedrain:Shannon 6.01 4.44 -2.91 14.43 1.00 3451 2810
## Ninitial16:StageWithoutseedrain:Shannon -3.39 6.73 -17.07 9.20 1.00 3293 2920
## Ninitial32:StageWithoutseedrain:Shannon -5.54 9.23 -23.25 13.63 1.00 3467 2659
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 6.73 0.19 6.37 7.14 1.00 5704 2920
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2343049 0.02458069 0.1866864 0.2831217
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 31.42 2.02 27.35 35.36 1.00 1831 2150
## StageWithoutseedrain 27.84 2.31 23.02 32.36 1.00 1658 1561
## StageWithseedrain:Shannon 3.71 1.21 1.35 6.10 1.00 1803 2134
## StageWithoutseedrain:Shannon 3.83 1.71 0.51 7.30 1.00 1625 1523
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 13.35 0.34 12.69 14.03 1.00 2756 1912
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.04788901 0.01431746 0.02216935 0.07802648
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 43.41 7.59 28.51 58.15 1.00 4655 3109
## Ninitial4:StageWithseedrain 48.90 6.71 35.63 62.21 1.00 4653 2803
## Ninitial8:StageWithseedrain 47.19 7.43 32.42 61.98 1.00 5048 2504
## Ninitial16:StageWithseedrain 37.01 9.66 18.04 55.13 1.00 4752 2817
## Ninitial32:StageWithseedrain 17.96 13.71 -9.50 44.75 1.00 4552 3196
## Ninitial2:StageWithoutseedrain 44.95 9.69 26.33 63.88 1.00 5308 2896
## Ninitial4:StageWithoutseedrain 42.39 6.11 30.34 54.14 1.00 5263 3186
## Ninitial8:StageWithoutseedrain 52.12 6.49 39.10 64.50 1.00 5084 2966
## Ninitial16:StageWithoutseedrain 18.26 6.38 5.82 30.71 1.00 4856 3189
## Ninitial32:StageWithoutseedrain 10.41 5.95 -1.15 22.12 1.00 5449 2661
## Ninitial2:StageWithseedrain:Shannon -7.98 6.32 -20.26 4.49 1.00 4679 2863
## Ninitial4:StageWithseedrain:Shannon -8.49 5.01 -18.40 1.61 1.00 4664 3033
## Ninitial8:StageWithseedrain:Shannon -4.54 4.76 -14.13 4.75 1.00 5035 2286
## Ninitial16:StageWithseedrain:Shannon 1.80 4.86 -7.40 11.30 1.00 4803 2797
## Ninitial32:StageWithseedrain:Shannon 8.89 5.54 -1.93 20.17 1.00 4511 3306
## Ninitial2:StageWithoutseedrain:Shannon -13.47 9.02 -30.99 3.68 1.00 5385 3007
## Ninitial4:StageWithoutseedrain:Shannon -7.43 5.00 -16.93 2.33 1.00 5173 2955
## Ninitial8:StageWithoutseedrain:Shannon -12.27 4.87 -21.61 -2.60 1.00 5114 2940
## Ninitial16:StageWithoutseedrain:Shannon 11.01 4.05 3.22 18.92 1.00 4895 3028
## Ninitial32:StageWithoutseedrain:Shannon 12.62 3.32 6.15 19.07 1.00 5619 2642
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 11.83 0.33 11.22 12.51 1.00 9144 2956
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1468118 0.02249241 0.1039205 0.1911467
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 91.64 0.47 90.71 92.56 1.00 1944 1826
## StageWithoutseedrain 71.71 0.56 70.60 72.85 1.00 1898 2008
## StageWithseedrain:Shannon 0.56 0.18 0.21 0.91 1.00 1899 1773
## StageWithoutseedrain:Shannon 0.18 0.31 -0.43 0.79 1.00 1958 2087
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 3.56 0.09 3.39 3.75 1.00 2755 2165
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.8971855 0.002267047 0.8922541 0.9011551
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 98.58 3.75 91.17 105.85 1.00 4014 3000
## Ninitial4:StageWithseedrain 99.64 4.32 91.13 108.00 1.00 3891 3161
## Ninitial8:StageWithseedrain 94.86 6.31 82.29 107.20 1.00 4063 3178
## Ninitial16:StageWithseedrain 96.25 10.48 75.76 116.92 1.00 3846 2966
## Ninitial32:StageWithseedrain 92.48 15.41 62.57 123.22 1.00 3970 2828
## Ninitial2:StageWithoutseedrain 67.66 2.05 63.58 71.63 1.00 3997 2927
## Ninitial4:StageWithoutseedrain 68.56 1.94 64.72 72.31 1.00 3412 2810
## Ninitial8:StageWithoutseedrain 74.03 2.16 69.79 78.27 1.00 3846 2955
## Ninitial16:StageWithoutseedrain 66.26 1.97 62.36 70.06 1.00 4136 2993
## Ninitial32:StageWithoutseedrain 71.14 2.75 65.76 76.41 1.00 4301 2691
## Ninitial2:StageWithseedrain:Shannon -4.08 2.47 -8.85 0.72 1.00 4025 2926
## Ninitial4:StageWithseedrain:Shannon -2.67 2.04 -6.63 1.35 1.00 3889 3068
## Ninitial8:StageWithseedrain:Shannon -0.34 2.35 -4.95 4.34 1.00 4055 3234
## Ninitial16:StageWithseedrain:Shannon -0.86 3.13 -7.05 5.25 1.00 3857 3035
## Ninitial32:StageWithseedrain:Shannon 0.18 3.92 -7.66 7.78 1.00 3962 2849
## Ninitial2:StageWithoutseedrain:Shannon 3.78 1.51 0.93 6.83 1.00 3936 3016
## Ninitial4:StageWithoutseedrain:Shannon 3.06 1.21 0.71 5.39 1.00 3487 2969
## Ninitial8:StageWithoutseedrain:Shannon -0.81 1.08 -2.93 1.37 1.00 3838 2763
## Ninitial16:StageWithoutseedrain:Shannon 2.68 0.93 0.82 4.49 1.00 4025 2990
## Ninitial32:StageWithoutseedrain:Shannon -0.06 1.11 -2.21 2.14 1.00 4290 2693
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 2.97 0.08 2.81 3.15 1.00 8566 2769
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.927432 0.001620812 0.923994 0.9302469
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Stage + Stage:Shannon
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain 63.06 1.52 60.22 66.03 1.00 1820 1865
## StageWithoutseedrain 43.42 1.93 39.61 47.32 1.00 1708 1886
## StageWithseedrain:Shannon 2.02 0.64 0.77 3.24 1.00 1809 1908
## StageWithoutseedrain:Shannon 4.59 1.21 2.15 6.98 1.00 1680 1857
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.80 0.28 10.28 11.36 1.00 2844 2214
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.3980418 0.02115576 0.3547234 0.4374282
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: productivity ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon
## Data: d_ (Number of observations: 640)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain 101.16 6.53 88.43 114.02 1.00 3930 3068
## Ninitial4:StageWithseedrain 95.49 8.93 78.25 113.11 1.00 4222 2459
## Ninitial8:StageWithseedrain 105.29 11.22 82.97 126.94 1.00 4199 2928
## Ninitial16:StageWithseedrain 103.48 15.79 72.81 134.61 1.00 3865 3118
## Ninitial32:StageWithseedrain 131.59 20.80 91.09 171.76 1.00 3806 3389
## Ninitial2:StageWithoutseedrain 50.33 4.25 41.89 58.59 1.00 4700 2948
## Ninitial4:StageWithoutseedrain 66.10 4.38 57.57 74.71 1.00 4729 2883
## Ninitial8:StageWithoutseedrain 63.96 4.60 54.93 73.12 1.00 3837 3192
## Ninitial16:StageWithoutseedrain 79.92 6.48 67.36 92.40 1.00 4083 3022
## Ninitial32:StageWithoutseedrain 77.88 9.93 58.27 97.24 1.00 3971 2813
## Ninitial2:StageWithseedrain:Shannon -24.61 4.49 -33.41 -15.65 1.00 3925 3144
## Ninitial4:StageWithseedrain:Shannon -14.02 4.67 -23.13 -4.99 1.00 4243 2648
## Ninitial8:StageWithseedrain:Shannon -15.26 4.62 -24.20 -6.12 1.00 4184 2808
## Ninitial16:StageWithseedrain:Shannon -11.49 5.35 -21.99 -1.17 1.00 3852 3122
## Ninitial32:StageWithseedrain:Shannon -17.73 5.99 -29.25 -6.14 1.00 3820 3344
## Ninitial2:StageWithoutseedrain:Shannon -3.34 3.37 -9.87 3.44 1.00 4693 2966
## Ninitial4:StageWithoutseedrain:Shannon -10.32 3.00 -16.14 -4.54 1.00 4750 2913
## Ninitial8:StageWithoutseedrain:Shannon -7.10 2.73 -12.58 -1.80 1.00 3807 3160
## Ninitial16:StageWithoutseedrain:Shannon -13.87 3.47 -20.63 -7.02 1.00 4069 2967
## Ninitial32:StageWithoutseedrain:Shannon -11.23 4.94 -20.82 -1.43 1.00 3991 2899
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 7.97 0.23 7.54 8.44 1.00 8386 2429
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5791532 0.01669405 0.5439361 0.6099267
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.